Cross-Domain Pretraining for Steady-State Neural CFD Surrogates

Anthony Zhou; Amir Barati Farimani; Shirley Ho; Rudy Morel

cs.LGOct 7, 2026

Cross-Domain Pretraining for Steady-State Neural CFD Surrogates

Authors: Anthony Zhou, Amir Barati Farimani, Shirley Ho, Rudy Morel

Organizations: Carnegie Mellon University · New York University · Polymathic AI · Princeton University · Flatiron Institute, Center for Computational Astrophysics · Flatiron Institute, Center for Computational Mathematics · Flatiron Institute, Scientific Computing Core

Abstract

Neural surrogates for computational fluid dynamics (CFD) have the potential to greatly enhance engineering innovation through accelerating simulation. However, the primary limitation for neural surrogates is the lack of generalization to geometries and applications beyond the training set, which is significant given the diversity of engineering scenarios. Currently, this is addressed by generating a new dataset for a specific application; however, this requires running costly numerical solvers. In this work, we take a step toward addressing this by studying neural surrogates trained across different geometries, boundary conditions, and fidelities. We find that cross-domain pretraining improves zero- and few-shot performance on held-out datasets relative to both training from scratch and transferring from domain-specific experts. In particular, finetuning a pretrained, cross-domain model can achieve 2-3x lower errors at the same sample size and use 8x fewer samples to achieve the same error, compared to training from scratch. This benefit is architecture agnostic and improves with model size and pretraining dataset diversity. Furthermore, we study how and why cross-domain pretraining works in CFD surrogates, and find that simply pooling steady-state datasets is both sufficient and effective. Given the high cost of generating CFD data, leveraging existing datasets through cross-domain pretraining will likely be a valuable strategy as future surrogates expand to tackle new problems and use cases.

Figures & tables

Figure 1: After pretraining a joint model across $6$ CFD datasets, we evaluate its zero/few-shot performance on $4$ held out datasets (see Table 1 ).

Name	Geometry	Closure	Solver	# Samples	# Cells	Re #	Inlet (m/s)	AoA ( ∘ )
DrivAerNet++	DrivAer Car	RANS $\kappa\text{-}\omega$	OpenFOAM	8000	24M	8e6	30	-
DrivAerML	DrivAer Car	SA-DDES	OpenFOAM	500	160M	7.2e6	38.9	-
WindsorML	Windsor Body	WMLES	Volcano	355	280M	2.9e6	40	-
AhmedML ∗	Ahmed Body	SA-DDES	OpenFOAM	500	20M	7.7e5	1	-
SHIFT-Sub ∗	Submarine	RANS	Luminary	100	32M	3.2e7	5	-
Emmi Wing	Tapered Wings	RANS SA	OpenFOAM	30,000	3.3M	5-20e6	[150, 300]	[-10, 10]

Table 1: Datasets considered in this work, roughly split into automotive (top) and aerospace (bottom) applications, spanning a variety of geometries, fidelities and physical regimes. Datasets with an asterisk ∗ are held-out for downstream evaluation.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.68	0.54	0.49	0.34	0.64	0.39	0.33	0.33	0.68	0.23	0.23	0.23	0.81	0.63	0.64	0.63
Domain Experts
DrivAerNet++	0.53	0.18	0.15	0.12	0.61	0.22	0.13	0.10	0.73	0.15	0.11	0.11	1.10	0.53	0.47	0.46
Emmi-Wing	0.73	0.21	0.21	0.19	0.66	0.24	0.19	0.16	0.61	0.14	0.12	0.12	1.10	0.52	0.48	0.49

Table 2: Validation error on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

Figure 2: Zero-shot predictions ( $C_{p}$ ) of pretrained models on a SHIFT-CCA geometry. Expert models can overfit to their pretraining sets, while the Joint model better predicts new geometries.

Figure 3: Validation loss vs. number of finetuning samples, plotted for the pretrained joint model and a model initialized from scratch. Two finetuning budgets are used (400/4000 steps). Note both axes are log scale.

Figure 4: Error of joint models on held-out datasets, after using $S=0$ or $S=8$ finetuning samples. Left: Joint models are pretrained with different numbers of pretraining datasets (2/4/6), with the model size held constant ( $N_{p}=\text{32M}$ ). Right: Joint models are pretrained at varying model sizes (9/32/121M), with the size of the pretraining dataset held constant ( $N_{d}=\text{6}$ ).

Model	DrivAerNet	Emmi-Wing	WindsorML	Double-Delta	DrivAerML	SuperWing
DrivAerNet	0.143	0.855	0.470	0.973	0.401	0.830
Emmi-Wing	0.809	0.030	0.617	1.239	0.830	0.742
WindsorML	0.678	0.747	0.067	0.956	0.701	0.736
Double-Delta	0.766	0.657	0.587	0.057	0.781	0.668
DrivAerML	0.421	0.786	0.458	0.966	0.048	0.803
SuperWing	0.973	0.648	0.705	1.255	1.006	0.047

Table 3: Validation error of expert and joint models across the pretraining datasets. Joint models are trained at different pretraining dataset sizes

N_{d}=2,4,6

. Cells are colored by error magnitude.

Appendix figures & tables35 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: The geometry (left), coefficient of pressure $C_{p}$ (center), and x-component of the skin friction coefficient $C_{f_{x}}$ (right) visualized for a single sample in all datasets. From top to bottom: DrivAerNet, DrivAerML, WindsorML, AhmedML, SHIFT-Submarine, Emmi Wing, SuperWing, HiLiftAeroML, Double Delta, SHIFT-CCA.

Figure 7: The volumetric pressure (shown as the coefficient of pressure $C_{p}$ ) and non-dimensional volumetric velocity magnitude $|\vec{u}|/U_{\infty}$ , plotted for a single sample in all datasets. From top to bottom: DrivAerNet, DrivAerML, WindsorML, AhmedML, SHIFT-Submarine, Emmi Wing, SuperWing, HiLiftAeroML, Double Delta, SHIFT-CCA.

SMART		SMART-IC		AB-UPT		Transolver++
Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
Model size	32.0M	Model size	32.4M	Model size	36.1M	Model size	29.1M
Hidden dim	384	Hidden dim	384	Hidden dim	384	Hidden dim	384
Heads	8	Heads	8	Heads	8	Heads	8
Enc./dec. blocks	12	Enc./dec. blocks	8	Shared blocks	ppscscs	Layers	12
Num latents	4096	Num latents	4096	Supernodes	1024	Slices	32

Appendix

Table 5: Model hyperparameters. All models use a hidden width of 384 and are matched at roughly 30M parameters. The optimizer is also constant across all models (Adam (

\beta_{1}{=}0.9

\beta_{2}{=}0.999

), learning rate

10^{-4}

, StepLR

\gamma{=}0.99

every 1k steps)

Figure 8: Schematic of the SMART model variants. Cross-attention is routed so that the encoder and decoder branches can share weights (link symbols). Gray arrows indicate subsampling. Simulation parameters $\theta$ enter through Feature-wise Linear Modulation (FiLM) layers ( Perez et al., 2017 )

Figure 9: Schematics of AB-UPT and Transolver. Gray arrows indicate subsampling. For clarity, branches and weight sharing in AB-UPT, and slicing/unslicing in Transolver, are not shown. Simulation parameters $\theta$ enter through Adaptive LayerNorm (AdaLN) layers.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.68	0.35	0.33	0.25	0.65	0.32	0.28	0.30	0.69	0.28	0.27	0.26	0.82	0.63	0.63	0.61
Domain Experts
DrivAerNet++	0.61	0.24	0.21	0.16	0.62	0.28	0.23	0.17	0.71	0.19	0.16	0.16	0.90	0.53	0.48	0.48
Emmi-Wing	0.74	0.24	0.22	0.19	0.68	0.29	0.24	0.22	0.70	0.20	0.18	0.17	1.02	0.60	0.54	0.53

Appendix

Table 6: Validation error for AB-UPT on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random Init.	0.68	0.36	0.34	0.30	0.64	0.31	0.29	0.27	0.68	0.22	0.21	0.20	0.81	0.61	0.59	0.58
Domain Experts
DrivAerNet++	0.59	0.27	0.27	0.19	0.57	0.28	0.24	0.22	0.79	0.21	0.19	0.18	0.97	0.56	0.46	0.51
Emmi-Wing	0.70	0.27	0.24	0.21	0.66	0.29	0.24	0.22	0.65	0.21	0.18	0.18	1.01	0.58	0.50	0.49

Appendix

Table 7: Validation error for Transolver++ on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

	Model	DrivAerNet	DrivAerML	Emmi-Wing	WindsorML	SuperWing	Double-Delta	Mean
Expert	SMART	0.142	0.047	0.030	0.065	0.047	0.055	0.064
	AB-UPT	0.148	0.057	0.030	0.068	0.058	0.064	0.071
	Transolver++	0.150	0.067	0.029	0.074	0.063	0.071	0.076
Joint	SMART	0.173	0.134	0.041	0.074	0.082	0.090	0.099
	AB-UPT	0.177	0.148	0.041	0.080	0.089	0.097	0.105
	Transolver++	0.184	0.166	0.044	0.097	0.107	0.124	0.120

Appendix

Table 8: SMART, AB-UPT, and Transolver++ models, evaluated on the pretraining datasets. Note that each cell in the Expert block is a different model (for example, the DrivAerNet SMART expert), while each row in the Joint block is a single model evaluated across all datasets.

Quantity	Dataset-specific	Global
Pressure coefficient	$(C_{p}-\mu_{d}^{(C_{p})})/\sigma_{d}^{(C_{p})}$	$C_{p}/f_{C_{p}}$ , $f_{C_{p}}=0.33$
Skin friction coefficient	$(C_{f}-\mu_{d}^{(C_{f})})/\sigma_{d}^{(C_{f})}$	$C_{f}/f_{C_{f}}$ , $f_{C_{f}}=0.002$
Velocity	$(u^{}-\mu_{d}^{(u^{})})/\sigma_{d}^{(u^{})}$ , $u^{}=\vec{u}/U_{\infty}$	$u^{\prime}/f_{u^{\prime}}$ , $u^{\prime}=(\vec{u}-\vec{U}_{\infty})/U_{\infty}$ , $f_{u^{\prime}}=0.2$
Simulation parameters	$(\theta-\theta_{\min})/(\theta_{\max}-\theta_{\min})$	$(\theta-\theta_{\min})/(\theta_{\max}-\theta_{\min})$
Geometry and queries	$(G-\bar{c}_{G})/s_{d}$ , $(x-\bar{c}_{G})/s_{d}$	$(G-\bar{c}_{G})/s_{d}$ , $(x-\bar{c}_{G})/s_{d}$
Estimated from simulation	$\mu_{d},\sigma_{d}$ for every field	None

Appendix

Table 9: Comparison of dataset-specific and global normalization. The subscript

d

denotes a quantity computed separately for each dataset.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random Init.	0.82	0.42	0.40	0.33	0.82	0.38	0.31	0.30	0.85	0.22	0.21	0.21	0.90	0.65	0.65	0.64
Domain Experts
DrivAerNet++	0.53	0.15	0.13	0.12	0.89	0.19	0.12	0.09	0.89	0.13	0.11	0.10	0.94	0.55	0.47	0.46
Emmi-Wing	0.71	0.20	0.20	0.19	0.67	0.24	0.17	0.15	0.68	0.14	0.12	0.12	0.82	0.54	0.46	0.46

Appendix

Table 10: Validation error for SMART (with global normalization) on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

Model	Normalization	DrivAerNet	DrivAerML	Emmi-Wing	WindsorML	SuperWing	Double-Delta	Mean
Expert	Dataset-specific	0.142	0.047	0.030	0.067	0.048	0.057	0.065
	Global	0.142	0.046	0.030	0.067	0.047	0.058	0.065
Joint	Dataset-specific	0.173	0.134	0.043	0.074	0.082	0.090	0.099
	Global	0.172	0.132	0.041	0.074	0.079	0.091	0.098

Appendix

Table 11: Comparison of dataset-specific and global normalization during pretraining. Losses are computed in the de-normalized space (raw

C_{p},C_{f},u^{*}

). Note that each cell in the Expert block is a different model, while each row in the Joint block is a single model evaluated across all datasets.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random Init.	0.77	0.57	0.54	0.40	0.83	0.53	0.49	0.47	0.87	0.53	0.45	0.46	0.87	0.76	0.75	0.76
Domain Experts
DrivAerNet++	0.57	0.17	0.14	0.12	0.64	0.31	0.23	0.19	0.88	0.34	0.26	0.24	0.97	0.65	0.57	0.49
Emmi-Wing	0.81	0.22	0.21	0.19	0.77	0.38	0.31	0.27	0.77	0.28	0.26	0.26	0.98	0.62	0.54	0.57

Appendix

Table 12: Validation error for SMART (evaluated on simulation mesh) on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

Model	DrivAerNet	Emmi-Wing	WindsorML	Double-Delta	DrivAerML	SuperWing
Joint (Base)	0.174	0.043	0.075	0.091	0.134	0.082
Joint (121M)	0.150	0.039	0.072	0.070	0.074	0.074
Joint (6xSteps)	0.150	0.029	0.069	0.063	0.073	0.041

Appendix

Table 13: Validation error of the joint model across the pretraining datasets, either with an increased model size (121M) or increased gradient budget (6xSteps).

Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Joint (Base)	0.31	0.12	0.10	0.09	0.52	0.14	0.10	0.09	0.52	0.11	0.10	0.10	0.90	0.46	0.34	0.33
Joint (121M)	0.31	0.12	0.09	0.08	0.53	0.16	0.11	0.08	0.55	0.09	0.08	0.08	0.92	0.45	0.32	0.28
Joint (6xSteps)	0.32	0.12	0.10	0.08	0.61	0.16	0.10	0.09	0.55	0.10	0.09	0.09	1.11	0.44	0.32	0.30

Appendix

Table 14: Validation error on four held-out target datasets. Each row is a different joint model, either with increased model size (121M) or increased gradient budget (6xSteps). Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation.

Model	DrivAerNet	Emmi-Wing	WindsorML	Double-Delta	DrivAerML	SuperWing
Joint (Base)	0.174	0.043	0.075	0.091	0.134	0.082
Joint (IC)	0.176	0.075	0.076	0.098	0.144	0.098

Appendix

Table 15: Pretraining validation error of the joint base model (SMART) and the in-context model (SMART-IC) on each dataset. SMART-IC additionally receives a solved context simulation from the same dataset as the query. Providing context does not reduce the error on any dataset.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.82	0.74	0.65	0.25	0.88	0.53	0.39	0.37	0.97	0.18	0.16	0.16	0.96	0.68	0.69	0.68
Domain Experts
DrivAerNet++	0.65	0.21	0.17	0.14	0.87	0.36	0.20	0.13	0.96	0.13	0.08	0.07	1.08	0.59	0.53	0.52
Emmi-Wing	0.85	0.24	0.22	0.20	0.90	0.40	0.30	0.22	0.89	0.12	0.10	0.10	1.12	0.58	0.51	0.52

Appendix

Table 16: Validation error in surface pressure coefficient

C_{p}

on four held-out target datasets. Rows indicate the model initialization; single-domain experts are identified by their pretraining dataset. Columns give the number of fine-tuning samples, with 0 denoting zero-shot evaluation. Green and red indicate the best and worst single-domain experts, respectively, and bold indicates the lowest overall error.

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.65	0.61	0.60	0.49	0.41	0.35	0.34	0.32	0.41	0.30	0.31	0.31	0.77	0.73	0.74	0.73
Domain Experts
DrivAerNet++	0.64	0.17	0.14	0.12	0.45	0.19	0.13	0.12	0.44	0.20	0.18	0.18	0.86	0.62	0.56	0.55
Emmi-Wing	0.75	0.19	0.19	0.18	0.49	0.19	0.16	0.15	0.39	0.19	0.18	0.18	1.39	0.63	0.59	0.61

Appendix

Table 17: Validation error in surface skin-friction coefficient

C_{f}

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.98	0.57	0.47	0.36	1.00	0.42	0.33	0.36	0.99	0.21	0.21	0.21	0.98	0.62	0.63	0.63
Domain Experts
DrivAerNet++	0.63	0.24	0.21	0.18	0.85	0.24	0.13	0.09	1.14	0.14	0.09	0.08	1.48	0.53	0.46	0.46
Emmi-Wing	1.01	0.28	0.30	0.27	1.01	0.29	0.21	0.18	0.91	0.13	0.10	0.10	1.13	0.51	0.46	0.47

Appendix

Table 18: Validation error in volume pressure coefficient

C_{p}

No Pretraining
Held-out Datasets:	AhmedML				SHIFT-Sub				SHIFT-CCA				HiLiftAeroML
$\#$ FT Samples:	0	2	4	8	0	2	4	8	0	2	4	8	0	2	4	8
Random init.	0.26	0.26	0.25	0.25	0.26	0.25	0.25	0.25	0.35	0.24	0.25	0.25	0.53	0.49	0.50	0.49
Domain Experts
DrivAerNet++	0.19	0.09	0.07	0.06	0.27	0.08	0.07	0.06	0.39	0.12	0.11	0.11	0.97	0.38	0.33	0.33
Emmi-Wing	0.29	0.12	0.13	0.12	0.24	0.09	0.09	0.08	0.25	0.12	0.11	0.12	0.77	0.37	0.35	0.36

Appendix

Table 19: Validation error in volume velocity

\mathbf{u}^{*}

Figure 10: Model predictions of surface $C_{p}$ on AhmedML.

Figure 11: Model predictions of surface $C_{f_{x}}$ on AhmedML.

Figure 12: Model predictions of volume $C_{p}$ on AhmedML.

Figure 13: Model predictions of volume $|\mathbf{u}^{*}|$ on AhmedML.

Figure 14: Model predictions of surface $C_{p}$ on SHIFT-Submarine.

Figure 15: Model predictions of surface $C_{f_{x}}$ on SHIFT-Submarine.

Figure 16: Model predictions of volume $C_{p}$ on SHIFT-Submarine.

Figure 17: Model predictions of volume $|\mathbf{u}^{*}|$ on SHIFT-Submarine.

Figure 18: Model predictions of surface $C_{p}$ on SHIFT-CCA.

Figure 19: Model predictions of surface $C_{f_{x}}$ on SHIFT-CCA.

Figure 20: Model predictions of volume $C_{p}$ on SHIFT-CCA.

Figure 21: Model predictions of volume $|\mathbf{u}^{*}|$ on SHIFT-CCA

Figure 22: Model predictions of surface $C_{p}$ on HiLiftAeroML.

Figure 23: Model predictions of surface $C_{f_{x}}$ on HiLiftAeroML.

Figure 24: Model predictions of volume $C_{p}$ on HiLiftAeroML.

Figure 25: Model predictions of volume $|\mathbf{u}^{*}|$ on HiLiftAeroML

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Pretraining a neural PDE surrogate can reduce the amount of new CFD data needed when geometry or modeled physics changes. However, it remains unclear how different components of distribution shift affect this benefit. We pretrain a surrogate on 254,909 RANS solutions from one airfoil family and fine-tune it on a new family under two target settings with matched freestream ranges: the same Spalart-Allmaras (SA) modeling and SA with added

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